the interplay of antiferromagnetism and …qpt.physics.harvard.edu/talks/cifar12.pdfthe interplay of...

The interplay of antiferromagnetism and superconductivity:

new results and open questions

HARVARDsachdev.physics.harvard.edu

Subir Sachdev

Saturday, October 20, 12

Max Metlitski

HARVARD

Erez Berg

Matthias Punk


1. Sign-problem free quantum Monte Carlo for the onset of antiferromagnetism in metals

2. Hole-doped cuprates: Where are the electron pockets in the Brillouin zone ?

Is the pseudogap state an exotic metal with Fermi pockets which violate the Luttinger relation ? (Such a violation requires emergent gauge excitations: an example of such a metal is the Z2-FL* state)

Outline


BaFe2(As1�x

Px

)2

K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).

Resistivity⇠ ⇢0 +ATn


BaFe2(As1�x

Px

)2

K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).


Nd2�xCexCuO4

T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,

M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,

Phys. Rev. Lett. 103, 157002 (2009).

Increasing SDW order

s


Fermi surface+antiferromagnetism

The electron spin polarization obeys�

⌃S(r, �)⇥

= ⌃⇥(r, �)eiK·r

where K is the ordering wavevector.

+

Metal with “large” Fermi surface



h~'i = 0

Antiferromagnetic metal with electron and hole pockets


h~'i 6= 0

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Increasing interaction



“Yukawa” coupling between fermions andantiferromagnetic order:

�2 ⇠ U , the Hubbard repulsion

S =

Zd2rd⌧ [Lc + L' + Lc']

Lc = c†a"(�ir)ca

L' =1

2(r'↵)

2 +r

2'2↵ +

u

4

�'2↵

�2

Lc' = �'↵ eiK·r c†a �↵ab cb.

Theory for onset of antiferromagnetism


Fermi surfaces translated by K = (�,�).



“Hot” spots



Electron and hole pockets in

antiferromagnetic phase with h~'i 6= 0



Pairing “glue” from antiferromagnetic fluctuations

V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983)D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986)

K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010)


Unconventional pairing at and near hot spots

��

Dc†k�c

†�k⇥

E= ��⇥�(cos kx � cos ky)


LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuationsFluctuating, paired Fermi

pockets

QCP for the onset of SDW order is actually within a superconductor


QuantumCritical


“Hot” spotsSaturday, October 20, 12

Low energy theory for critical point near hot spotsSaturday, October 20, 12

v1 v2

�2 fermionsoccupied

�1 fermionsoccupied

Theory has fermions 1,2 (with Fermi velocities v1,2)

and boson order parameter ~',interacting with coupling �

kx

ky




L = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

+1

2(rr ~')

2 +s

2~'2 +

u

4~'4

��~' ·⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).Saturday, October 20, 12



L = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

+1

2(rr ~')

2 +s

2~'2 +

u

4~'4

��~' ·⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘

Note fermion spectrum has lines of zero energy

excitations in momentum space.

If fermions are replaced by massless Dirac

fermions with points of zero energy excitations,

then critical theory is well-understood




L = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

+1

2(rr ~')

2 +s

2~'2 +

u

4~'4

��~' ·⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘

M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010)S.A. Hartnoll, D.M. Hofman, M. A. Metlitski and S. Sachdev, Phys. Rev. B 84, 125115 (2011)




L = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

+1

2(rr ~')

2 +s

2~'2 +

u

4~'4

��~' ·⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘

M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010)S.A. Hartnoll, D.M. Hofman, M. A. Metlitski and S. Sachdev, Phys. Rev. B 84, 125115 (2011)

Theory flows to

strong-coupling in d = 2.


v1 v2



kx

ky

To faithfully realize low energy theory in quantum Monte Carlo,

we need a UV completion in which Fermi lines don’t end

and all weights are positive.


K

Hot spots in a single band model

QMC for the onset of antiferromagnetism


K

Hot spots in a two band model


E. Berg, M. Metlitski, and

S. Sachdev, arXiv:1206.0742


Faithful realization

of the generic

universal low

energy theory for the onset

of antiferro-

magnetism.

K






K






Sign problem is absent as long as K connects

hotspots in distinct bands

K






K





Particle-hole or point-group

symmetries or commensurate densities not

required !

Sign problem is absent as long as K connects

hotspots in distinct bands



Electrons with dispersion "

k

interacting with fluctuations of the

antiferromagnetic order parameter ~'.

Z =

ZDc

↵

D~' exp (�S)

S =

Zd⌧

X

k

c

†k↵

✓@

@⌧

� "

k

◆c

k↵

+

Zd⌧d

2x

1

2

(rx

~')

2+

r

2

~'

2+ . . .

�

� �

Zd⌧

X

i

~'

i

· (�1)

xic

†i↵

~�

↵�

c

i�


Electrons with dispersions "(x)k

and "

(y)k

interacting with fluctuations of theantiferromagnetic order parameter ~'.

Z =

ZDc

(x)↵

Dc

(y)↵

D~' exp (�S)

S =

Zd⌧

X

k

c

(x)†k↵

✓@

@⌧

� "

(x)k

◆c

(x)k↵

+

Zd⌧

X

k

c

(y)†k↵

✓@

@⌧

� "

(y)k

◆c

(y)k↵

+

Zd⌧d

2x

1

2(r

x

~')2+

r

2~'

2 + . . .

�

� �

Zd⌧

X

i

~'

i

· (�1)xic

(x)†i↵

~�

↵�

c

(y)i�

+H.c.






and "

(y)k


Z =

ZDc

(x)↵

Dc

(y)↵

D~' exp (�S)

S =

Zd⌧

X

k

c

(x)†k↵

✓@

@⌧

� "

(x)k

◆c

(x)k↵

+

Zd⌧

X

k

c

(y)†k↵

✓@

@⌧

� "

(y)k

◆c

(y)k↵

+

Zd⌧d

2x

1

2(r

x

~')2+

r

2~'

2 + . . .

�

� �

Zd⌧

X

i

~'

i

· (�1)xic

(x)†i↵

~�

↵�

c

(y)i�

+H.c.


No sign problem !





and "

(y)k


Z =

ZDc

(x)↵

Dc

(y)↵

D~' exp (�S)

S =

Zd⌧

X

k

c

(x)†k↵

✓@

@⌧

� "

(x)k

◆c

(x)k↵

+

Zd⌧

X

k

c

(y)†k↵

✓@

@⌧

� "

(y)k

◆c

(y)k↵

+

Zd⌧d

2x

1

2(r

x

~')2+

r

2~'

2 + . . .

�

� �

Zd⌧

X

i

~'

i

· (�1)xic

(x)†i↵

~�

↵�

c

(y)i�

+H.c.


Applies without changes to the

microscopic band structure in the

iron-based superconductors





and "

(y)k


Z =

ZDc

(x)↵

Dc

(y)↵

D~' exp (�S)

S =

Zd⌧

X

k

c

(x)†k↵

✓@

@⌧

� "

(x)k

◆c

(x)k↵

+

Zd⌧

X

k

c

(y)†k↵

✓@

@⌧

� "

(y)k

◆c

(y)k↵

+

Zd⌧d

2x

1

2(r

x

~')2+

r

2~'

2 + . . .

�

� �

Zd⌧

X

i

~'

i

· (�1)xic

(x)†i↵

~�

↵�

c

(y)i�

+H.c.




Can integrate out ~' toobtain an extendedHubbard model. The

interactions in this modelonly couple electrons in

separate bands.


K






!1

!0.5

0

0.5

1k y/! K

a)


Center Brillouin zone at (π,π,)




−1 0 1−1

−0.5

0

0.5

1

kx/π

k y/π

−1 0 1−1

−0.5

0

0.5

1

k y/π kx/π

K

a) b)


Move one of the Fermi surface by (π,π,)




!1

!0.5

0

0.5

1k y/!

a)


Now hot spots are at Fermi surface intersections




−1 0 1−1

−0.5

0

0.5

1

kx/π

k y/π

−1 0 1−1

−0.5

0

0.5

1k y/π

kx/π

K

a) b)


Expected Fermi surfaces in the AFM ordered phase




E. Berg, M. Metlitski, and S. Sachdev, arXiv:1206.0742


−1 0 1−1

−0.5

0

0.5

1r = −0.5

kx/π

k y/π

−1 0 1−1

−0.5

0

0.5

1

kx/π

r = 0

−1 0 1−1

−0.5

0

0.5

1

kx/π

r = 0.5

0.5

1

1.5

Electron occupation number nk

as a function of the tuning parameter r



−0.5 0 0.5 1

0

0.1

0.2

0.3

0.4

rBi

nder

cum

ulan

t−0.5 0 0.5 10

0.2

0.4

0.6

r

χ φ/(L

2 β)

L=8L=10L=12L=14

a) b)

AF susceptibility, �', and Binder cumulant

as a function of the tuning parameter r




−2 −1 0 1 2 3−2

0

2

4

6

8

10 x 10−4

r

P ±(xmax

)L = 10

L = 14

L = 12

rc

↑

P+

_

_P_

|

s/d pairing amplitudes P+/P�as a function of the tuning parameter r




−2 −1 0 1 2 3−2

0

2

4

6

8

10 x 10−4

r

P ±(xmax

)L = 10

L = 14

L = 12

rc

↑

P+

_

_P_

|

Notice shift between the position of the QCP in the superconductor, and the position of maximum pairing.

This is found in numerous experiments.



1. Sign-problem free quantum Monte Carlo for the onset of antiferromagnetism in metals

2. Hole-doped cuprates: Where are the electron pockets in the Brillouin zone ?

Is the pseudogap state an exotic metal with Fermi pockets which violate the Luttinger relation ? (Such a violation requires emergent gauge excitations: an example of such a metal is the Z2-FL* state)

Outline


Hole-doped

Electron-doped

?


35

Consistent with antinodal pseudogap (at least at low fields)

35

C. C. Holmes et al. Phys. Rev. Lett. 71, 1645 (1993)

Peets D C… Damascelli A New J. Phys. 9 28 (2007)

YBCO6+x

Pseudogap state has no low energy fermions in the antinodal region


48

Resonant x-ray scattering Surprise – wavevectors 3.3a, biaxial

48

E ≃ 931 eV Q = (0.31, 0) and (0, 0.31) Correlation length ξ≈ 60Å

Ghiringelli et al. (http://dx.doi.org/10.1126/science.1223532)

YBa2Cu3O6.6

QCDW 2QSDW


53 53

E ≃ 100 keV YBa2Cu3O6.67 (ortho-VIII)

Correlation length ξ≈ 98Å (at 2K and 17T) qCDW= q1=(δ1,0,0.5) and q2=(0,δ2,0.5),where δ1 ≈ 0.30 and δ2≈0.31.

Hard x-ray scattering in agreement with results from resonant

x-ray scattering

J. Chang et al. arXiv: 1206.4333


40

Nodal pocket with electron-like direction of cyclotron motion

40

N. Harrison & SES Phys. Rev. Lett. 106, 226402 (2011) S. E. Sebastian et al., Rep. Prog. Physics (in press, 2012)

40

Nodal pocket with electron-like direction of cyclotron motion

40

N. Harrison & SES Phys. Rev. Lett. 106, 226402 (2011) S. E. Sebastian et al., Rep. Prog. Physics (in press, 2012)


51

Or potentially, a wavevector connecting tips of arcs

51 S. E. Sebastian et al., Rep. Prog. Physics (in press, 2012)

In reality, such an electron pocket is a reconstruction of a parent state which has no Fermi surface in the anti-nodal region. So if we

accept this picture, the parent state is very exotic ! A metal with Fermi surfaces only near the nodes, and no broken

symmetry, which violates the Luttinger relation. Saturday, October 20, 12


Quantum phase transition with Fermi surface reconstruction

h~'i = 0

Metal with electron and hole pockets


h~'i 6= 0





h~'i 6= 0 h~'i = 0

Separating onset of SDW orderand Fermi surface reconstruction





h~'i 6= 0 h~'i = 0

Fractionalized Fermi liquid (FL*) phasewith no symmetry

breaking and “small” Fermi surface

h~'i = 0


Electron and/or hole Fermi pockets form in “local” SDW order, but quantum fluctuations destroy long-range

SDW order

T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003)Saturday, October 20, 12

Heavy Fermi liquid with “large” Fermi

surface of hydridized f and

c-conduction electrons

Magnetic order and the heavy Fermi liquid in the Kondo lattice

h~'i = 0

Magnetic Metal: f-electron moments

and c-conduction electron

Fermi surface

h~'i 6= 0

f

c

f+c


Separating onset of SDW order and the heavy Fermi liquid in the Kondo lattice



Fermi surface

h~'i 6= 0

f

c




h~'i = 0

f+c





Fermi surface

h~'i 6= 0

f

c




h~'i = 0

f+c

c

f

Conduction electronFermi surface

andspin-liquid of f-electrons

h~'i = 0





Fermi surface

h~'i 6= 0

f

c




h~'i = 0

f+c

c

f



h~'i = 0


f- sp

in:

loca

lized

delo

caliz

ed

Fractionalized Fermi liquid (FL*)

AFM Metal

large Fermi surface heavy Fermi liquid

YbAlB4

K

Q

Kc

Qc

QC1

QC2

YbRh2Si2Yb(Rh0.94Ir0.06)2Si2

YbAgGe

YbRh2(Si0.95Ge0.05)2

Yb(Rh0.93Co0.07)2Si2

QTC

Kondo screened paramagnet

spin

liqu

id

YbIr2Si2

CeCu2Si2

Experimental perpective on same phase diagrams of Kondo lattice

J. Custers, P. Gegenwart, C. Geibel, F. Steglich,

P. Coleman, and S. Paschen, Phys. Rev. Lett.

104, 186402 (2010)





h~'i 6= 0 h~'i = 0



h~'i = 0


Electron and/or hole Fermi pockets form in “local” SDW order, but quantum fluctuations destroy long-range

SDW order


0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

k_x

k_y

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

k_x

k_y

Hole pocket of a Z2-FL* phasein a single-band t-J model

M. Punk and S. Sachdev, Phys. Rev. B 85, 195123 (2012)Saturday, October 20, 12

Characteristics of FL* phase

• Fermi surface volume does not countall electrons.

• Such a phase must have neutral S = 1/2 ex-citations (”spinons”), and collective spinlessgauge excitations (“topological” order).

• These topological excitations are needed toaccount for the deficit in the Fermi surfacevolume, in M. Oshikawa’s proof of theLuttinger theorem.


Solved sign-problem for generic universal theory for the onset of antiferromagnetism in

two-dimensional metals. Good prospects for studying non-Fermi liquid

physics at non-zero temperature

Conclusions


Pseudo-gap phase of hole-doped cuprates asa “fractionalized Fermi liquid”:

Hole pockets which violate the Luttinger theorem, in a phase with “topological” excitations.

Conclusions


the interplay of antiferromagnetism and …qpt.physics.harvard.edu/talks/cifar12.pdfthe interplay of...

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